We study a gravity-driven viscous flow coating a vertical cylindrical fibre. The destabilisation of a draining liquid column into a downward moving train of beads has been linked to the conjunction of the Rayleigh-Plateau and Kapitza instabilities in the limit of small Bond numbers Bo. Here, we focus on quasi-inertialess flows (large Ohnesorge number Oh) and conduct a linear stability analysis on a unidirectional flow along a rigid eccentric fibre for intermediate to large Bo. We show the existence of two unstable modes, namely pearl and whirl modes. The pearl mode depicts asymmetric beads, similar to that of the Rayleigh-Plateau instability, whereas a single helix forms along the axis in the whirl mode instability. The geometric and hydrodynamic thresholds of the whirl mode instability are investigated, and phase diagrams showing the transition thresholds between different regimes are presented. Additionally, an energy analysis is carried out to elucidate the whirl formation mechanism. This analysis reveals that despite the unfavourable capillary energy cost, the asymmetric interface shear distribution, caused by the fibre eccentricity, has the potential to sustain a whirling interface. In general, small fibre radius and large eccentricity tend to foster the whirl mode instability, while reducing Bo tends to favour the dominance of the pearl mode instability. Finally, we compare the predictions of our model with the results of some illustrative experiments, using highly viscous silicone oils flowing down fibres. Whirling structures are observed for the first time, and the measured wavenumbers match our stability analysis prediction.